Einstein metrics on homogeneous spaces H × H / ΔK

Abstract

Given any compact homogeneous space H/K with H simple, we consider the new space M = H × H/ΔK, where ΔK denotes diagonal embedding, and study the existence, classification and stability of H × H-invariant Einstein metrics on M, as a first step into the largely unexplored case of homogeneous spaces of compact non-simple Lie groups. We find unstable Einstein metrics on M for most spaces H/K such that their standard metric is Einstein (e.g., isotropy irreducible) and the Killing form of k is a multiple of the Killing form of h (e.g., K simple), a class which contains 17 families and 50 individual examples. A complete classification is obtained in the case when H/K is an irreducible symmetric space and K is simple. We also study the behavior of the scalar curvature function on the space of all normal metrics on M = H × H/ΔK (none of which is Einstein), obtaining that the standard metric is a global minimum.